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Modulational Instability in Equations of KdV Type

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Book cover New Approaches to Nonlinear Waves

Part of the book series: Lecture Notes in Physics ((LNP,volume 908))

Abstract

It is a matter of experience that nonlinear waves in a dispersive medium, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics—the amplitude, the phase, the wave number, etc.—slowly vary in large space and time scales. In the 1960s, Whitham developed an asymptotic (WKB) method to study the effects of small “modulations” on nonlinear dispersive waves. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham’s formal theory. We discuss some recent advances in the mathematical understanding of the dynamics, in particular, the instability, of slowly modulated waves for equations of KdV type.

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Notes

  1. 1.

    Note that \(\mathcal{O}(1)\) terms cancel out thanks to the definition of \((a_{0},E_{0},c_{0})\), and hence the leading order term is \(\mathcal{O}(\varepsilon )\).

  2. 2.

    The requirement that the perturbation be integrable is not the only choice possible. Other natural candidates of classes of perturbations include periodic perturbations with fundamental period nT for some \(n = 1, 2, 3,\ldots\). While such periodic classes of perturbations are natural from a variational viewpoint, they may impose artificial constraints on the physical problem. As we will see below, the class of localized, i.e. integrable on the line, perturbations includes information about all quasi-periodic perturbations.

  3. 3.

    Here we ignore issues of well-posedness of Eq. (4.1) to such initial data.

  4. 4.

    In general, spectral stability does not imply linear stability, as is familiar from the ODE theory. Nevertheless, spectral instability often does imply linear (and nonlinear) instability.

  5. 5.

    Here, we take the convention that the left hand side is zero when k = 0.

  6. 6.

    Indeed, this formula for D was derived in [12] using a direct spectral perturbation expansion, a method equivalent to that described in Sect. 4.4.2.

  7. 7.

    Following [41], the projections here are slightly re-scaled from those of Sect. 4.4. Here we use \(M_{\xi }:= \left [\langle \psi _{i},L_{\xi }\phi _{j}\rangle /\langle \psi _{i},\phi _{j}\rangle _{L_{\mathrm{per}}^{2}([0,T])}\right ]_{i,j=1,2,3}\) and similarly for the projection of the identity.

  8. 8.

    The condition α > 1∕2 is an artifact of the corresponding existence theory. See [45] for details.

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Acknowledgements

JCB is supported by the National Science Foundation grant DMS-1211364. VMH is supported by the National Science Foundation grant CAREER DMS-1352597 and an Alfred P. Sloan Foundation Fellowship. MAJ is supported by the National Science Foundation grant DMS-1211183.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Jared C. Bronski & Vera Mikyoung Hur

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Mathew A. Johnson

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  1. Jared C. Bronski
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  1. Institute for Analysis, Johannes Kepler University, Linz, Austria

    Elena Tobisch

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Bronski, J.C., Hur, V.M., Johnson, M.A. (2016). Modulational Instability in Equations of KdV Type. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_4

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